After playing for a while, I managed to beat the game in just 3 jumps.
First of all, what's a jump?
Pressing jump on the ground is jump. Grace jumping is also a jump.
Grace jumping from a dash is not a jump though, so you can do that.
Wall jumping is not a jump.
And jumping on a spike is not a jump.
With these rules, you can beat the game with just 3 jumps.
This is not a tutorial on how to do it, but if people want to see how, I can make a video on it.
Edit: I lowered it down to 1 jump, so yay have fun
Edit2: Video guide: